Prove Theorem 4 18(c) Combine the proofs of parts (a) and (b) and see the fourth Remark following Theorem 3 9 (page 175) a For any positive integer n, n is an eigenvalue of A n with corresponding eigenvector x b If A is invertible, then 1 is an eigenvalue of A 1 with corresponding eigenvector x c I...

Using induction and the proofs of 418a and b we will prove Theorem 418c For any integernifAxx thenA ...

Prove Theorem 4.18(c). [Combine the proofs of parts (a) and (b) and see the fourth Remark following

Question:

Prove Theorem 4.18(c). [Combine the proofs of parts (a) and (b) and see the fourth Remark following Theorem 3.9 (page 175).]

a. For any positive integer n, λⁿ is an eigenvalue of Aⁿ with corresponding eigenvector x.
b. If A is invertible, then 1/λ is an eigenvalue of A¹ with corresponding eigenvector x.
c. If A is invertible, then for any integer n, λⁿ is an eigenvalue of Aⁿ with corresponding eigenvector x.